On the differential geometric Schottky problem

利用Weierstrass表示定理，研究了n维欧氏空间中复正交群在单连通极小曲面上的作用，并确定了极小曲面作为该作用的极限.出现极小曲面是Enneper极小曲面的推广。如果给定的极小曲面不是全纯的，那么这个作用使极小曲面不稳定。我们考虑了三维欧氏空间中一条曲线的框架（如Nitsch框架）的Plateau问题。本文研究了极小曲面在标架变形下的分叉问题，并通过计算机模拟给出了突变现象的几何可视化。这与微分几何Schottky问题（研究Siegel上半空间中黎曼矩阵空间的弯曲性）有很深的关系。我们考虑环面变形空间余切丛中的突变集。然后我们得到了一个具有复结构的Lagrange锥，并推广了这一构造。我们考虑了欧氏空间中具有较高余维的极小曲面系的Dirichlet问题（给定边界映射）。对于边界映射，存在解和无解。我们可以认为，在边界映射变形的情况下，会出现具有奇异性的解。事实上，Lawson和Osserman得到了一个作为极小锥的解。我们推广了它们的构造，并给出了许多解作为极小锥。此外，我们还看到极小锥具有形变空间，因此研究包含我们结果的极小曲面的形变是很重要的。

We consider the action of the complex orthogonal group on a simply connected minimal surface in the n-dimensional Euclidean space by using the Weierstrass representation theorem and decide the minimal surface appearing as limits of this action. An appearing minimal surface is a generalization of the Enneper minimal surface. If the given minimal surface is not holomorphic, then this action makes the minimal surface to be unstable. The reason is that the bifurcation (Catastrophe phenomena) occurs.We consider the Plateau problem for the frame (a curve in the 3 dimensional Euclidean space) like the Nitsch frame. We study the bifurcation of a minimal surface by a deformation of the frame and see the geometric visualization of the catastrophe phenomenon by a computer simulation.We consider a deformation of a minimal surface in a torus by a deformation of the torus. This is deeply relative to the Differential geometric Schottky problem (investigate the curvedness of the space of Riemann matrices in the Siegel upper half space). We consider the catastrophe set in the cotangent bundle of the space of the deformation of tori. Then we obtain a Lagrangian cone with a complex structure and generalize the construction.We consider a minimal surface system in the Euclidean space with higher co-dimension for the Dirichlet problem (a given boundary map). For the boundary map, there exists a solution and no solution.. We may consider that the solution with singularity appears under the deformation of the boundary map. In fact, Lawson and Osserman obtain a solution as a minimal cone. We generalize their construction and give many solutions as minimal cones. Moreover we see that a minimal cone have the deformation space.Hence it is important to study the deformation of minimal surfaces which contains our results.